I am struggling with exercise 9.8 of Paulsen's book "Completely bounded maps and operator algebras". It says:
Prove that a finite direct sum of operators $T_1\oplus \dots T_n$ is similar to a contraction if and only if each operator in the direct sum is similar to a contraction. Moreover, prove that $\inf \{\|S\|\|S^{-1}\|: \|S^{-1}(T_1\oplus \cdots\oplus T_n)S\| \leq 1\}$ is achieved by an operator $S$ that is itself a direct sum.
I can prove the first part: By a theorem of the book: We call $\mathcal{H}:= \mathcal{H}_1 \oplus \dots \oplus \mathcal{H}_n$ with $T_i$ acting in $\mathcal{H}_i$. By theorem 9.11 of the book an operator $T \in \mathcal{H}$ is similar to a contraction if and only if $\rho: P(\mathbb{D})\to B(\mathcal{H})$ of polynomials defined in de disk (seen as operator subspace of the continuous functions in the disk) $\rho(p) = p(T)$ is completely bounded. Moreover, $\|\rho\|_{cb} = \inf \{\|S\|\|S^{-1}\|: \|S^{-1}(T_1\oplus \cdots\oplus T_n)S\| \leq 1\}$ and the infimum is achieved.
Hence we can define $\rho_i:P(\mathbb{D}) \to B(\mathcal{H}_i)$, $\rho_i(p)=p(T_i)$. With this at hand, suppose that $\rho$ is a contraction. then
$\infty>\|\rho_T\| = \sup_{\|p\|=1}\|p(T)\|\geq \sup_{\|p\|=1}\|p_i(T_i)\| = \|\rho_i\|. $
and in a similar vein for all the amplifications, so $\rho_i$ is completely bounded for all $i$. By the theorem above, $T_i$ is similar to a contraction. Reciprocally, if $T_i$ is similar to a contraction for all $i$, there exist similarities $S_i$ such that $\|S_iT_iS_i^{-1}\|_{B(\mathcal{H}_i)}\leq 1$. Then, the similarity $S\in B(\mathcal{H})$ $S(x_1,\dots,x_n) = (S_1x_1,\dots,S_nx_n)$ verifies that $\|STS^{-1}\| = \|S_1T_1T_1^{-1}\oplus\dots\oplus S_nT_nT_n^{-1}\| = \max\{\|S_iT_iT_i^{-1}\|_{B(\mathcal{H}_i)}: 1\leq i\leq n\}\leq 1,$ and hence, $T$ is similiar to a contraction.
However, I cannot deal with the second statement: I don't know why the infimum would be achieved in a diagonal operator. I understand that there exists $S_k$ such that $ \|\rho\|_{cb} = \|S_k\|\|S_k^{-1}\|$ with $S_k$ as above but I don't know how to extend $S_k$ to an invertible and diagonal operator in $\mathcal{H}$.
Thank you in advice.